Five-Minute Check (over Lesson 7–1)

CCSS

Then/Now

New Vocabulary

Key Concept: Property of Equality for Exponential Functions

Example 1: Solve Exponential Equations

Example 2: Real-World Example: Write an Exponential Function

Key Concept: Compound Interest

Example 3: Compound Interest

Key Concept: Property of Inequality for Exponential Functions

Example 4: Solve Exponential Inequalities

Over Lesson 7–1

A. D = {x | x < 0}, R = {all real numbers}

B. D = {x | x > 0}, R = {all real numbers}

C. D = {all real numbers}, R = {y | y < 0}

D. D = {all real numbers}, R = {y | y > 0}

State the domain and range of y = –3(2)x.

Over Lesson 7–1

A. D = {x | x < 0}, R = {all real numbers}

B. D = {x | x > 0}, R = {all real numbers}

C. D = {all real numbers}, R = {y | y < 0}

D. D = {all real numbers}, R = {y | y > 0}

State the domain and range of

Over Lesson 7–1

A. $3619.12

B. $4112.64

C. $8882.36

D. $9375.88

The function P(t) = 12,995(0.88)t gives the value of a type of car after t years. Find the value of the car after 10 years.

Over Lesson 7–1

A. P(t) = 25(1.40)t

B. P(t) = 25(1.60)t

C. P(t) = 10t

D. P(t) = 15t

The number of bees in a hive is growing exponentially at a rate of 40% per day. The hive begins with 25 bees. Which function models the population of the hive after t days?

Content Standards

A.CED.1 Create equations and inequalities in one variable and use them to solve problems.

F.LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Mathematical Practices

2 Reason abstractly and quantitatively.

You graphed exponential functions.

• Solve exponential equations.

• Solve exponential inequalities.

• exponential equation

• compound interest

• exponential inequality

Solve Exponential Equations

A. Solve the equation 3x = 94.

3x = 94 Original equation

3x = (32)4 Rewrite 9 as 32.

3x = 38 Power of a Power

x = 8 Property of Equality for Exponential FunctionsAnswer: x = 8

Solve Exponential Equations

B. Solve the equation 25x = 42x – 1.

25x= 42x – 1 Original equation

25x= (22)2x – 1

Rewrite 4 as 22.

25x= 24x – 2

Power of a Power

5x= 4x – 2Property of Equality for Exponential Functions

x = –2 Subtract 4x from each side.

Answer: x = –2

A. 3

B. 9

C. 18

D. 27

A. Solve the equation 4x = 643.

A. 1

B. 2

C. 4

D. 5

B. Solve the equation 32x = 95x – 4.

Write an Exponential Function

A. POPULATION In 2000, the population of Phoenix was 1,321,045. By 2007, it was estimated at 1,512,986. Write an exponential function that could be used to model the population of Phoenix. Write x in terms of the numbers of years since 2000.

At the beginning of the timeline in 2000, x is 0 and the population is 1,321,045. Thus, the y-intercept, and the value of a, is 1,321,045.

When x = 7, the population is 1,512,986. Substitute these values into an exponential function to determine the value of b.

Write an Exponential Function

y = ab

x Exponential function

1,512,986 = 1,321,045 ● b7 Replace x with 7, y with 1,512,986, and a with 1,321,045.

1.145 ≈ b7 Divide each side by1,321,045.

Answer: An equation that models the number of years is y = 1,321,045(1.0196)x.

Take the 7th root ofeach side.

1.0196 ≈ b Use a calculator.

Write an Exponential Function

B. POPULATION In 2000, the population of Phoenix was 1,321,045. By 2007, it was estimated at 1,512,986. Predict the population of Phoenix in 2013.

y = 1,321,045(1.0196)x Modeling equation

y = 1,321,045(1.0196)13 Replace x with 13.

y ≈ 1,700,221 Use a calculator.

Answer: The population will be about 1,700,221.

A. y = 9,426(1.0963)x – 7

B. y = 1.0963(9,426)x

C. y = 9,426(x)1.0963

D. y = 9,426(1.0963)x

A. POPULATION In 2000, the population of the town of Tisdale was 9,426. By 2007, it was estimated at 17,942. Write an exponential function that could be used to model the population of Tisdale. Write x in terms of the numbers of years since 2000.

A. 28,411

B. 30,462

C. 32,534

D. 34,833

B. POPULATION In 2000, the population of the town of Tisdale was 9,426. By 2007, it was estimated at 17,942. Predict the population of Tisdale in 2012.

Compound Interest

An investment account pays 5.4% annual interest compounded quarterly. If $4000 is placed in this account, find the balance after 8 years.

Understand

Find the balance of the account after 8 years.

Plan

Use the compound interest formula.

P = 4000, r = 0.054, n = 4, and t = 8

Compound Interest Formula

Compound Interest

Solve

P = 4000, r = 0.054, n = 4, and t = 8

Use a calculator.

Answer: The balance in the account after 8 yearswill be $6143.56.

Compound Interest

Check

Graph the corresponding equation y = 4000(1.0135)4t. Use the CALC: value to find y when x = 8.

The y-value 6143.6 is very close to 6143.56, so the answer is reasonable.

A. $6810.53

B. $7420.65

C. $7960.43

D. $8134.22

An investment account pays 4.6% annual interest compounded quarterly. If $6050 is placed in this account, find the balance after 6 years.

Solve Exponential Inequalities

Original equation

Property of Inequality for Exponential Functions

Subtract 3 from each side.

Solve Exponential Inequalities

Divide each side by –2 and reverse the inequality symbol.

Answer:

A. x < 9

B. x > 3

C. x < 3

D. x > 6